Adjoint pseudospectral least-squares methods for an elliptic boundary value problem

The adjoint approach for Legendre pseudospectral least-squares methods is presented by adopting the adjoint first-order systems developed in [Z. Cai, T. Manteuffel, S. McCormick, J. Ruge, First-order system LL∗ (FOSLL*): scalar elliptic partial differential equations, SIAM J. Numer. Anal. 39 (2001) 1418–1445]. The discrete adjoint least-squares functional on a polynomial space using Legendre–Gauss–Lobatto points and weights is shown to be equivalent to H1 norm. The spectral convergence is also provided with several numerical results.